I Have Mac OS X Lion 10.7.5 (11G63). How high can I upgrade my software?

I'm guessing at everything because I don't even know what model Mac you have (iMac, Mini, MacBook, etc.)

But if you have a 2010 Mac, you should be able to upgrade to High Sierra. (It still depends on whether it is a mid or late year model). If not you can upgrade to El Capitan.

But here's the thing, you should have 6 to 8 GB RAM for it to run smoothly, less than that I wouldn't bother.

Look here, did you read this in the link I provided?

Can your Mac use Sierra?

macOS Sierra requires any of the following Mac models and versions of macOS. It also requires at least 2GB of memory and 8.8GB of storage space.

• iMac models from late 2009 or later
• MacBook models from late 2009 or later
• MacBook Pro models from mid 2010 or later
• MacBook Air models from late 2010 or later
• Mac mini models from mid 2010 or later
• Mac Pro models from mid 2010 or later

Need to know the specifics. Model and year (early/late). Apple menu > About this Mac.

See the Support Topic I linked to which will spell out how far you can go.

You can upgrade to High Sierra.

It depends on the age of your Mac.

How to download macOS Sierra - Apple Support